3.5 Eve’s Uncertainty Is Non-increasing Under Symmetrization
49
3.5 Eve’s Uncertainty Is Non-increasing Under
Symmetrization
Part of the proof in this Section is inspired by [38].
In computing the secret key rate of the BB84 protocol [5] in Sect. 3.2, we argue
that w.l.o.g. the state ρ AB distributed to Alice and Bob by Eve is replaced by (3.13):
˜
ρ AB =
1
4
[ρ AB + (Z ⊗ Z )ρ AB (Z ⊗ Z ) + (X ⊗ X )ρ AB (X ⊗ X )
+(Y ⊗ Y )ρ AB (Y ⊗ Y )] .
(3.33)
This scenario can be viewed as Eve preparing one of the four states
ρ AB , (Z ⊗ Z ) ρ AB (Z ⊗ Z ), (X ⊗ X ) ρ AB (X ⊗ X ), (Y ⊗ Y ) ρ AB (Y ⊗ Y )
(3.34)
depending on the outcome t = 1, 2, 3, 4 of a random variable stored in the register
T , which Eve is aware of. Since Eve holds the purifying system E of every state in
(3.34), the state prepared by Eve is:
˜
ρ AB E T =
1
4
t
|φ
t
AB E φ
t
AB E | ⊗ |tt | T ,
(3.35)
where {|φ
t
AB E }
4
t=1 are pure states. Finally, we assume that Eve holds the purifying
system T
of the state in (3.35). Thus the global state is pure and reads:
|φ AB E T T =
1
2
t
|φ
t
AB E ⊗ |t T ⊗ |t T .
(3.36)
Note that (3.36) is a purification of (3.35), where both registers T and T
are held
by Eve. The above argument holds only if it’s not disadvantageous for Eve. In
other words, Eve’s uncertainty on Alice’s key, quantified by the conditional entropy
H (R A |E), must be non-increasing. Therefore, we must verify that:
H (R A |E) ρ ≥ H (R A |E tot ) ˜
ρ ,
(3.37)
where Eve’s quantum system E tot = E T T
contains: the quantum side information
E, the outcome of the random variable T , and the purifying system T
.
Proof In order to prove (3.37), we start by using the strong subadditivity property
(c.f. Sect. 2.6):
H (R A |E tot ) ˜
ρ ≤ H (R A |E T ) ˜
ρ
(3.38)